Application of the Simplified
Hirota (Homogenization) Method
to a (3+1)-Dimensional
Evolution Equation
for Deriving Multiple Soliton Solutions
Ünal Göktaş∗ and Willy Hereman◦
∗ Department of Computer Engineering
Turgut Özal University, Keçiören, Ankara, Turkey
ugoktas@turgutozal.edu.tr
◦ Department of Applied Mathematics and Statistics
Colorado School of Mines, Golden, Colorado, U.S.A.
whereman@mines.edu
The Ninth IMACS International Conference on
Nonlinear Evolution Equations and Wave Phenomena:
Computation and Theory
Athens, Georgia, U.S.A.
Thursday, April 2, 2015, 5:05 p.m.
Acknowledgements
Willy Malfliet (University of Antwerp, Belgium)
Frank Verheest (University of Ghent, Belgium)
Research supported in part by NSF
under Grant No. CCF-0830783
Outline
•
Motivation
•
The Method
•
Application to a (3+1)-Dimensional Evolution
Equation
•
Conclusion
Motivation
•
Hirota’s method can be used to find exact
solutions of nonlinear PDEs, provided these
equations can be brought into a bilinear form.
•
Finding bilinear forms for nonlinear PDEs, if they
exist at all, is highly nontrivial.
•
To circumvent this difficulty, we introduce a
simplified version of Hirota’s method, and use it
to construct solitary and soliton solutions.
•
Without bilinear forms, exact solutions can still be
constructed straightforwardly by solving a
perturbation scheme on the computer, using a
symbolic manipulation package.
Simplified Version of Hirota’s Method
To illustrate the homogenization method, the well
known Korteweg-de Vries (KdV) equation will be used;
ut + buux + uxxx = 0,
(1)
where b is any real constant.
For the KdV equation, the transformation
12fxx
12fx 2
12 ∂ 2
u=
−
=
ln f.
2
2
bf
bf
b ∂x
(2)
will allow one to homogenize the equations, once the
equation has been “homogenized” it can be readily
solved.
First, integrate the equation with respect to x as
many times as possible and ignore integration
“constants” (i.e., arbitrary functions of t). For the
KdV equation only one integration is possible, yielding
Z x
b 2
ut dx + u + u2x = 0.
(3)
2
Second, change of dependent variable according to
∂2
(f f2x − fx2 )
u(x, t) = K 2 ln f (x, t) = K
,
2
∂x
f
(4)
to transform (1) into a homogeneous equation in f and
its derivatives of as low a degree as possible. For the
KdV quation, setting K = 12
two terms vanish which
b
allows one to cancel a common factor f 2 . Hence, one
obtains a homogeneous equation of second degree,
2
= 0.
f [fxt + f4x ] − fx ft − 4fx f3x + 3f2x
(5)
The approach below works for “quadratic” equations
of the form
f L(f ) + N (f, f ) = 0,
(6)
where L denotes a linear differential operator and N is
a bilinear (quadratic) differential operator.
For the KdV equation,
L(f ) = fxt + f4x ,
(7)
N (f, g) = −fx gt − 4fx g3x + 3f2x g2x .
(8)
and
Third, seek a solution of the form
f (x, t) = 1 +
∞
X
n f (n) (x, t),
n=1
where serves as a bookkeeping parameter (not a
small quantity).
Substitute (9) into (6).
(9)
Use Cauchy’s product formula,



∞
∞
n−1
∞
X
X
X
X
r
s
n

ar  
bs  =
an−j bj ,
r=1
s=1
n=2
(10)
j=1
to group the terms in powers of . Equate to zero the
different powers of :
O(1 ) : L(f (1) ) = 0,
(11)
O(2 ) : L(f (2) ) = −N (f (1) , f (1) ),
..
.
(12)
O(n ) :
L(f (n) ) = −[
n−1
X
f (n−j) L(f (j) ) +
j=2
n−1
X
N (f (n−j) , f (j) )],
j=1
n ≥ 3.
(13)
The latter formula can be written succinctly as
L(f (n) ) = −
n−1
Xh
i
f (n−j) L(f (j) ) + N (f (n−j) , f (j) ) ,
n ≥ 3,
j=1
(14)
provided one uses L(f (1) ) = 0.
For example, application of (13) with n = 3 gives
h
i
L(f (3) ) = − f (1) L(f (2) ) + N (f (2) , f (1) ) + N (f (1) , f (2) ) .
(15)
The N-soliton solution of the KdV is then generated
from
f (1) =
N
X
i=1
exp(θi ) =
N
X
exp (ki x − ωi t + δi ),
(16)
i=1
where ki , ωi and δi are constants. Substitution of (16)
into (11) yields the dispersion relation P (ki , ωi ) = 0.
For the KdV equation,
P (ki , ωi ) = ki (ki 3 − ωi ),
i = 1, 2, ..., N.
(17)
The dispersion law is thus ωi = ki3 . Using (16), we
compute the right hand side of (12):
−
N
X
3ki kj2 (ki − kj ) exp(θi + θj ) =
i,j=1
X
3ki kj (ki − kj )2 exp(θi + θj ).
(18)
1≤i<j≤N
Observe that the terms in exp(2θi ) are missing, which
is typical for quadratic equations in f which admit
solitons.
Furthermore, (18) determines the form of f (2) :
X
(2)
f =
aij exp(θi + θj ) =
(19)
1≤i<j≤N
X
aij exp[(ki + kj )x − (ωi + ωj )t + (δi + δj )].
1≤i<j≤N
With (7), (17), and (19), the left hand side of (12) is
readily computed
X
L(f (2) ) =
P (ki + kj , ωi + ωj ) aij exp(θi + θj )
1≤i<j≤N
=
X
1≤i<j≤N
3ki kj (ki + kj )2 aij exp(θi + θj ).
(20)
Equating (18) and (20), we have
aij =
(ki − kj )2
2,
(ki + kj )
1 ≤ i < j ≤ N.
(21)
Proceeding in a similar way with (15) leads to the
explicit form of f (3) .
For example, for N = 3 we have
f (3) = b123 exp(θ1 + θ2 + θ3 )
= b123 exp[(k1 + k2 + k3 )x − (ω1 + ω2 + ω3 )t
+(δ1 + δ2 + δ3 )],
(22)
with
b123 = a12 a13 a23 =
(k1 − k2 )2 (k1 − k3 )2 (k2 − k3 )2
2
2
(k1 + k2 ) (k1 + k3 ) (k2 + k3 )
2.
(23)
For N = 3, one can show by the computation of the
rest of the scheme that f (n) = 0 for n > 3. Therefore
the expansion in (9) truncates, and
f = 1 + exp θ1 +exp θ2 +exp θ3 + a12 exp(θ1 +θ2 )
+a13 exp(θ1 +θ3 ) + a23 exp(θ2 +θ3 )
+b123 exp(θ1 +θ2 +θ3 ),
where we set = 1.
(24)
Note that f has no terms in exp(2θ1 ), exp(2θ2 ),
exp(2θ1 + θ2 ) and exp(2θ2 + θ1 ), . . .
Upon substitution of (24) into (4) the well-known
three-soliton solution of the KdV equation follows.
Conclusion
Hirota’s direct method for solving completely
integrable PDEs is algorithmic and implementable in
the language of any computer algebra system.
The main advantages of the simplified version of
Hirota’s method is that the bilinear representation for
the equation becomes superfluous.
Therefore, the method can be applied to equations for
which the bilinear form is not known or does not exist.
Software packages in Mathematica
Codes are available via the Internet:
URL: http://inside.mines.edu/∼whereman/
Thank You